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Journal of the London Mathematical Society Advance Access originally published online on January 29, 2007
Journal of the London Mathematical Society 2007 75(1):163-175; doi:10.1112/jlms/jdl021
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© 2007 London Mathematical Society

Freiman's theorem in an arbitrary abelian group

Ben Green

Centre for Mathematical Sciences
Wilberforce Road
Cambridge CB3 0WA
United Kingdom
b.j.green{at}dpmms.cam.ac.uk

Imre Z. Ruzsa

Alfréd Rényi Mathematical Institute
Hungarian Academy of Sciences
Budapest
Pf. 127, H-1364, Hungary
ruzsa{at}renyi.hu

A famous result of Freiman describes the structure of finite sets A {subseteq} Z with small doubling property. If |A + A| ≤ K|A|, then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.

2000 Mathematics Subject Classification 11P70, 11B99.

While this work was carried out, the first author was supported by a PIMS postdoctoral fellowship at the University of British Columbia, Vancouver, Canada.

Received May 23, 2005; revised February 13, 2006; published online January 29, 2007.


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